OPEN ACCESS
The paper is concerned with the approach developed within the ANR Project StaRAC, and it gives an overview of its main results. The objective was to reconsider the concept of stationarity so as to make it operational, allowing for both an interpretation relatively to an observation scale and the possibility of its testing thanks to the use of timefrequency surrogates, as well as to offer various extensions, especially beyond shift invariance.
Extended Abstract
As explained in Section 1, the word “stationarity” is ubiquitous in signal processing and data analysis but, often used in a loose sense, it may correspond to different qualities that are not necessarily captured by what is referred to as “stationarity” in textbooks. Classically (Loève, 1962 ; Doob, 1967), stationarity refers to stochastic processes and is defined as the invariance in time of statistical properties or, in other words, as the independence of those properties with respect to some absolute time. In practice, however, stationarity is commonly advocated in rather different contexts and/or with additional features that implicitly enter the picture.
Two examples are mentioned in Section 1.1 for supporting this claim. The first one concerns speech signals which, when considered at time scales of several seconds, are unanimously considered as “nonstationary”, and for which methods have been proposed for, e.g., their segmentation into “stationary” parts. The way those parts are referred to as stationary however differs significantly from the standard definition. On the one hand, some time scale is taken into account, what needs some accommodation with respect to the actual definition that extends over all times. On the other hand, an identification between stationarity and periodicity is routinely made (cf. voiced segments), which is another departure from the standard definition given in a stochastic framework. A second example is given by internet data, for which two distinct issues related to “stationarity” can be considered. If we first analyze such data at different time scales (thanks, e.g., to a wavelet decomposition), one can be interested as in the speech example in the time variability and its statistical significance. If we now look at the different scales jointly, we are faced with a different form of invariance that makes all of them look quite similar in terms of their variability (be it significant or not over the considered time scales) (Willinger et al., 1997; Park, Willinger, 2000; Loiseau et al., 2010). This invariance across scales is coined “self-similarity” (Embrechts, Maejima, 2002), but it clearly shares much with the idea of “stationarity”, provided that time shifts are replaced by dilations. These two examples may serve as an elementary motivation for the scientific objectives of StaRAC:
– propose and develop operational (i.e., interpretable, relative and testable) approaches to the concept of stationarity, with the purpose of filling existing gaps between theory and practice;
– develop new methods to test and measure departures from stationarity;
– extend the concept of stationarity to groups of transformations different from shifts.
Following Section 1.2 devoted to a state-of–the-art of existing approaches, Section 2 gives an overview of the framework in which the project has been developed. Section 2.1 first makes more precise the general concept of relative stationarity, whereas Section 2.2 details the way it is used in practice. The key point is to operate in a time-frequency (TF) domain and to explicitly introduce two time scales: a global one, fixed by the observation span, and a local one, aimed at evidencing variations within the former. From a practical point of view, the TF description relies on multi-taperspectrograms, and the assessment of stationarity, relatively to the observation scale, results from a comparison between local spectra and the global one obtained by marginalization.
In order to give a statistical significance to such comparisons, Section 2.3 then presents the original strategy that has been developed for characterizing the null hypothesis attached to stationarity. It is based on the use of a collection of surrogate data, constructed from the observation by randomizations of the phase of its spectrum (Theiler et al., 1992; Schreiber, Schmitz, 2000). It has been shown (Borgnat et al., 2010 ; Richard et al., 2010) that surrogates constructed this way are guaranteed to be stationary, paving the way for their use as stationarized versions of the data to be tested.
Given the outlined framework, the question of how to derive operational tests is the purpose of Section 3. Two main categories of tests have been envisioned (see (Borgnat et al., 2010) for a comprehensive presentation). The first one (Section 3.1) relies on “distances” between local and global spectra. A specific combination of the Kullback-Leibler divergence and the log-spectral deviation proved most useful, with a resulting test variable whose fluctuations follow approximately a Gamma distribution under the null hypothesis of stationarity (Xiao, Borgnat, Flandrin, 2007 ; Xiao et al., 2009). This allows not only for a specified confidence in the detection, but also for the obtention of by-products such as a degree and a typical scale of nonstationarity. A second approach (Section 3.2) considers the collection of surrogates as a learning set attached to the stationary hypothesis, with possible tests using techniques aimed at outlier detection, such as, e.g., one-class support vector machines or others (Xiao, Borgnat, Flandrin, Richard, 2007 ; Amoud et al., 2009b). This part is concluded (Section 3.3) by an example where detection is achieved in a specific feature space adapted to amplitude- and frequency-modulated waveforms. For a same degree of nonstationarity, this allows for a quantitative characterization of the type of this nonstationarity (Amoud et al., 2009a).
Connected approaches are briefly discussed in Section 4. Section 4.1 introduces 2D surrogates aimed at detecting either transients with unknown shape and location in the TF plane (Borgnat, Flandrin, 2008) or nonstationary cross-correlations in bivariate signals (Borgnat, Flandrin, 2009). Section 4.2 mentions possible 2D time-scale extensions based on wavelet decompositions in place of spectrograms, with the purpose of testing homogeneity in images (Flandrin, Borgnat, 2008). A greater attention is paid in Section 4.3 to a generalization of the concept of stationarity in the specific context of self-similarity. Going beyond the so-called Lamperti transformation (Flandrin et al., 2003) which connects self-similar processes with stationary ones, emphasis is put on a two-parameter perspective, based on the affine group, that permits to guarantee the additional property of stationary increments while dealing with finite size effects (Ramirez-Cobo et al., 2010). Finally, Section 4.4 addresses a problem that was not initially supposed to be dealt with in the project, namely a fresh perspective (based on EmpiricalMode Decomposition (Huang et al., 1998)) on the problem of decomposing a given observation in a trend and a fluctuation. In fact, this problem shares much with the general viewpoint developed previously for stationarity, where both the definition and the analysis are explicitly made dependent on the observation scale, and the newapproach is shown to compare favorably with existing ones (Moghtaderi, Flandrin, Borgnat, 2011 ; Moghtaderi, Borgnat, Flandrin, 2011).
RÉSUMÉ
Cet article présente l’approche suivie dans le projet ANR StaRAC et en résume les résultats principaux. L’objectif était de reconsidérer le concept de stationnarité dans le but de lui donner une forme opérationnelle, se prêtant à une interprétation relative à une échelle d’observation et permettant de le tester dans un sens statistique précis grâce à l’emploi de substituts temps-fréquence, ainsi que d’en fournir diverses extensions, en particulier au-delà de l’invariance en translation.
stationarity, test, time-frequency, spectral distances, learning, self-similarity.
MOTS-CLÉS
stationnarité, test, temps-fréquence, distances spectrales, apprentissage, autosimilarité.
Amblard P.-O., Borgnat P., Flandrin P. (2003). Stochastic processes with finite size scale invariance. In Proc. SPIE conf. on Noise and Fluctuations, vol. 5114, p. 161–172. Santa Fre, NM (USA).
Amoud H., Honeine P., Richard C., Borgnat P., Flandrin P. (2009a). Sur la caractérisation de non-stationnarité par la méthode des substituts. In 22e colloque sur le traitement du signal et des images. GRETSI-2009. Dijon (France).
Amoud H., Honeine P., Richard C., Borgnat P., Flandrin P. (2009b). Time-frequency learning machines for nonstationarity detection using surrogates. In IEEE Workshop on Statistical Signal Processing SSP-2009, p. 565–568. Cardiff (UK).
Aronszajn N. (1950). Theory of reproducing kernels. Trans. Amer. Math. Soc., vol. 68, p. 337–404.
Avaro O., Roux I. D. C., Hillion A. (2006). Generalized second-order invariance in texture modeling. Machine Graphics & Vision, vol. 15, no 1, p. 73–97.
Basseville M. (1989). Distances measures for signal processing and pattern recognition. Signal Processing, vol. 18, no 4, p. 349–369.
Bayram M., Baraniuk R. (2000). Multiple window time-varying spectrum estimation. In W. F. et al. (Ed.), Nonlinear and Nonstationary Signal Processing. Cambridge Univ. Press.
Borgnat P. (2006). Signal processing methods related to models of turbulence. In J. Fournier, J. Grimm, J. Leblond, J. Partington (Eds.), Harmonic analysis and rational approximation: Their rôles in signals, control, and dynamical systems theory, vol. 327, p. 277–301.
Borgnat P., Amblard P.-O., Flandrin P. (2005). Scale invariances and lamperti transformations for stochastic processes. J. Phys. A : Math. Gen., vol. 38, p. 2081–2101.
Borgnat P., Flandrin P. (2008). Time-frequency surrogates for nonstationary signal analysis. In 8th IMA International Conference on Mathematics in Signal Processing. Cirencester (UK).
Borgnat P., Flandrin P. (2009). Stationarization via surrogates. Journal of StatisticalMechanics: Theory and Experiment: Special issue UPoN 2008, p. P01001.
Borgnat P., Flandrin P., Amblard P.-O. (2002). Stochastic discrete scale invariance. IEEE Sig. Proc. Lett., vol. 9, no 6, p. 181–184.
Borgnat P., Flandrin P., Ferrari A., Richard C. (2011). Transitional surrogates. In IEEE Proc. ICASSP 2011, p. 3600–3603. Prague (République Tchèque).
Borgnat P., Flandrin P., Honeine P., Richard C., Xiao J. (2010). Testing stationarity with surrogates: A time-frequency approach. IEEE Trans. Signal Processing, vol. 58, no 7, p. 3459–3470.
Boser B., Guyon I., Vapnik V. (1992). A training algorithm for optimal margin classifiers. In Proc. Fifth Annual Workshop on Computational Learning Theory, p. 144–152.
Brcich R., Iskander D. (2006). Testing for stationarity in the frequency domain using a sphericity statistic. In IEEE Proc. ICASSP 2006, vol. III, p. 464–467. Toulouse (France).
Burnecki K., Maejima M., Weron A. (1997). The Lamperti transformation for self-similar processes. Yokohama Math. J., vol. 44, p. 25–42.
Clerc M., Mallat S. (2003). Estimating deformations of stationary processes. The Annals of Statistics, vol. 31, no 6, p. 1772–1821.
Dahlhaus R. (1996). Asymptotic statistical inference for nonstationary processes with evolutionary spectra. In Athens conference on applied probability and time series, vol. II, vol. 115, p. 145–159. Springer.
Doob J. (1967). Stochastic processes. Wiley and Sons.
Dubrulle B. (2000). Finite size scale invariance. Eur. Phys. J. B, vol. 14, p. 757–771.
Embrechts P., Maejima M. (2002). Selfsimilar processes. Princeton University Press.
Flandrin P. (2008). Time-frequency surrogates. Invited talk at the AMS Joint Mathematics Meeting, San Diego, CA (USA).
Flandrin P., Borgnat P. (2008). Revisiting and testing stationarity. J. Phys: Conf. Series "2008 Euro American Workshop on Information Optics WIO’08", vol. 139, p. 012004.
Flandrin P., Borgnat P., Amblard P.-O. (2003). From stationarity to self-similarity, and back: Variations on the Lamperti transformation. In G. Raganjaran, M. Ding (Eds.), Processes with long-range correlations: Theory and applications, vol. 621, p. 88–117. Springer-Verlag.
Flandrin P., Rilling G., Gonçalves P. (2004). Empirical mode decomposition as a filter bank. IEEE Sig. Proc. Lett., vol. 11, no 2, p. 112–114.
Fuentes M. (2005). A formal test for nonstationarity of spatial stochastic processes. J. Multiv. Anal., vol. 96, p. 30–54.
Gray H. L., Zhang N. F. (1988). On a class of nonstationary processes. Journal of Time Series Analysis, vol. 9, no 2, p. 133–154.
Hannan E. (1965). Group representations and applied probability. In J. Gani (Ed.), Supplementary review series in applied probability, vol. 3. Methuien & Co.
Hlawatsch F., Tauböck G., Twaroch T. (2002). Covariant time-frequency analysis. In L.Debnath (Ed.), Wavelets and signal processing. Boston (MA), Birkhäuser.
Hobijn B., Franses P., Ooms M. (2004). Generalization of the KPSS-test for stationarity. Statistica Neerlandica, vol. 58, p. 482–502.
Hodrick R., Prescott E. (1997). Postwar U.S. business cycles: An empirical investigation. Journal of Money, Credit, and Banking, vol. 29, no 1, p. 1–16.
Honeiné P., Richard C., Flandrin P. (2007). Time-frequency learning machines. IEEE Trans. on Signal Proc., vol. 55, no 7 (Part 2), p. 3930–3936.
Huang N. E., Shen Z., Long S. R.,Wu M. L., Shih H. H., Zheng Q. et al. (1998). The Empirical Mode Decomposition and Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. Roy. Soc. London A, vol. 454, p. 903–995.
Ijima H., Okui R., Ohsumi A. (2005). Detection of signals in nonstationary random noise via stationarization and stationarity test. In IEEE Workshop on Statistical Signal Processing SSP-2005. Bordeaux (France).
Kim S. J., Koh K., Boyd S., Gorinevsky D. (2009). l1 trend filtering. SIAM Review, vol. 51, no 2, p. 339–360.
Kwiatkowski D., Phllips P., Schmidt P., Shin Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root. J. of Econometrics, vol. 54, p. 159–178.
Lamperti J. (1962). Semi-stable stochastic processes. Trans. Amer. Math. Soc., vol. 104, p. 62-78.
Laurent H., Doncarli C. (1998). Stationarity index for abrupt changes detection in the timefrequency plane. IEEE Signal Proc. Lett., vol. 5, no 2, p. 43–45.
Lim S., Muniandy S. (2003). Generalized Ornstein-Uhlenbeck processes and associated selfsimilar processes. Journal of Physics A: Mathematical and General, vol. 36, p. 3961–3982.
Loiseau P., Gonçalves P., Dewaele G., Borgnat P., Abry P., Primet Vicat-Blanc P. (2010). Investigating self-similarity and heavy-tailed distributions on a large scale experimental facility. IEEE/ACM Trans. on Networking, vol. 18, no 4, p. 1261–1274.
Loève M. (1962). Probability theory. D. von Nostrand Comp.
Mallat S., Papanicolaou G., Zhang Z. (1998). Adaptive covariance estimation of locally stationary processes. Ann. of Stat., vol. 24, no 1, p. 1–47.
Martin W. (1984). Measuring the degree of non-stationarity by using the Wigner-Ville spectrum. In Proc. ieee icassp-84, p. 41B.3.1–41B.3.4. San Diego (CA).
Martin W., Flandrin P. (1985). Detection of changes of signal structure by using the Wigner-Ville spectrum. Signal Proc., vol. 8, p. 215–233.
Mika S., Rätsch G.,Weston J., Schölkopf B., Müller K. R. (1999). Fisher discriminant analysis with kernels. In Y. H. Hu, J. Larsen, E. Wilson, S. Douglas (Eds.), Advances in neural networks for signal processing, p. 41-48. San Mateo, CA, Morgan Kaufmann.
Moghtaderi A., Borgnat P., Flandrin P. (2011). Trend filtering: Empirical mode decompositions vs. ℓ1 and hodrick-prescott. Advances in Adaptive Data Analysis, vol. 3, no 1–2, p. 41–61.
Moghtaderi A., Flandrin P., Borgnat P. (2011). Trend filtering via empirical mode decompositions. Computational Statistics & Data Analysis. (In Press, Corrected Proof Available online)
Nuzman C., Poor V. (2000). Linear estimation of self-similar processes via Lamperti’s transformation. J. of Applied Probability, vol. 37, no 2, p. 429-452.
Park K., Willinger W. (2000). Self-similar network traffic and performance evaluation. New York, NY, USA, John Wiley & Sons, Inc.
Perrin O., Senoussi R. (1999). Reducing non-stationary stochastic processes to stationarity by a time deformation. Statistics & Probability Letters, vol. 43, p. 393–397.
Prichard D., Theiler J. (1994). Generating surrogate data for time series with several simultaneously measured variables. Physical Review Letters, vol. 73, no 7, p. 951–954.
Priestley M. B. (1988). Non-linear and Non-stationary Time Series Analysis. Academic Press.
Priestley M. B., Rao T. S. (1969). A test for non-stationarity of time-series. Journal of the Royal Statistical Society. Series B (Methodological), vol. 31, no 1, p. 140-149.
Ramirez-Cobo P., Amblard P.-O., Borgnat P., Flandrin P. (2010). Generalized selfsimilar sochastic processes with stationary increments. Rapport technique. GIPSA-lab& LP-ENS de Lyon.
Richard C., Ferrari A., Amoud H., Honeine P., Flandrin P., Borgnat P. (2010). Statistical hypothesis testing with time-frequency surrogates to check signal stationarity. In IEEE Proc. ICASSP 2011, p. 720–724. Dallas, TX (USA).
Schölkopf B., Smola A. J., Müller K. (1998). Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, vol. 10, no 5, p. 1299-1319.
Schreiber T., Schmitz A. (2000). Surrogate time series. Physica D, vol. 142, no 3-4, p. 346–382.
Silverman R. (1957). Locally stationary random processes. IRE Trans. on Info. Theory, vol. 3, p. 182–187.
Sornette D. (1998). Discrete scale invariance and complex dimensions. Physics Reports, vol. 297, p. 239–270.
Theiler J., Eubank S., Longtin A., Galdrikian B., Farmer J. D. (1992). Testing for nonlinearity in time series: the method of surrogate data. Physica D, vol. 58, no 1–4, p. 77–94.
Vapnik V. N. (1995). The nature of statistical learning theory. New York, Springer-Verlag.
Vaton S. (1998). A new test of stationarity and its application to teletraffic data. In Proceedings of ieee icassp-98, vol. 6, p. 3449–3452. Seattle (USA).
von Sachs R., Neumann M. (2000). A wavelet-based test for stationarity. J. Time Series Analysis, vol. 21, p. 597–613.
Willinger W., Taqqu M. S., Sherman R., Wilson D. V. (1997). Self-similarity through highvariability: statistical analysis of ethernet lan traffic at the source level. IEEE/ACM Trans. Netw., vol. 5, no 1, p. 71–86.
Xiao J., Borgnat P., Flandrin P. (2007). Sur un test temps-fréquence de stationnarité. In 21e Colloque sur le Traitement du Signal et des Images (gretsi), p. 957-960. Troyes, France, GRETSI.
Xiao J., Borgnat P., Flandrin P. (2007). Testing stationarity with time-frequency surrogates. In Proc. EUSIPCO 2007, p. 2020–2024. Poznan (PL).
Xiao J., Borgnat P., Flandrin P. (2009). Sur un test temps-fréquence de stationnarité. Traitement
du Signal, vol. 25, no 4, p. 357–366. ((in French, with extended English summary)) Xiao J., Borgnat P., Flandrin P., Richard C. (2007). Testing stationarity with surrogates – A one-class SVM approach. In IEEE Workshop on Statistical Signal Processing SSP-2007, p. 720–724. Madison (WI).
Xiao J., Flandrin P. (2007). Multitaper time-frequency reassignment for nonstatinary spectrum estimation and chirp enhancement. IEEE Trans. on Signal Proc., vol. 55, no 6 (Part 2), p. 2851–2860.
Yazici B., Kashyap R. L. (1997). A class of second-order stationary self-similar processes for 1/f phenomena. IEEE Trans. on Signal Proc., vol. 45, no 2, p. 396–410.